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[Enda Xiao](https://orcid.org/0000-0002-4372-1575), [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161)

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[High-throughput computational screening of Heusler compounds with phonon considerations for enhanced material discovery](https://mdr.nims.go.jp/datasets/eb5e8913-ba12-4715-adde-bf37ca10b692)

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High-throughput computational screening of Heusler compounds with phononconsiderations for enhanced material discoveryEnda Xiao∗Research Center for Magnetic and Spintronic Materials,National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, 305-0047, Ibaraki, JapanTerumasa Tadano†Research Center for Magnetic and Spintronic Materials,National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, 305-0047, Ibaraki, Japan andDigital Transformation Initiative Center for Magnetic Materials (DXMag),National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, 305-0047, Ibaraki, Japan(Dated: September 10, 2025)High-throughput (HTP) ab initio calculations are performed on 27,865 Heusler compositions,covering a broad range of regular, inverse, and half-Heusler compounds in both cubic and tetragonalphases. In addition to conventional stability metrics, such as formation energy, Hull distance, andmagnetic critical temperature Tc, phonon stability is assessed by systematically conducting ab initiophonon calculations for over 8,000 compounds. The performance of ab initio stability criteria issystematically assessed against 189 experimentally synthesized compounds, and magnetic criticaltemperature calculations are validated using 59 experimental data points. As a result, we identify631 stable compounds as promising candidates for further functional material exploration. Notably,47 low-moment ferrimagnets are identified, with their spin polarization and anomalous Hall/Nernstconductivity calculated to provide insights into potential applications in spintronics and energyharvesting. Furthermore, our analyses reveal linear relationship between Tc and magnetization in14 systems and correlations between stability and atomic properties such as atomic radius andionization energy. The regular/inverse structures preference in X2Y Z compound and tetragonaldistortion are also investigated for a broad Heusler family.I. INTRODUCTIONHeusler alloys are renowned for their exceptionalmagnetic and functional properties, including high-saturation magnetization, substantial magnetocrystallineanisotropy (MCA), elevated magnetic critical tempera-ture (Tc), significant magnetocaloric effects, and notablethermoelectric performance [1]. The unique combinationof these properties, along with the diverse compositionswithin the Heusler family, has prompted extensive high-throughput (HTP) studies to explore their potential forvarious applications [2–11]. Previous HTP investigationshave primarily relied on stable compounds in variousdatabases, such as the Heusler database at the Univer-sity of Alabama, the Open Quantum Materials Database(OQMD), and the AFLOW database [9, 12, 13]. How-ever, these databases are limited in the scope of theHeusler compound family, restricting the exploration offunctional material candidates.In HTP studies, the initial candidate pool is typicallynarrowed by assessing the thermodynamic stability ofcompounds. Thermodynamic stability is commonly eval-uated using formation energy and distance to the con-vex hull, which quantify stability relative to its decom-position into constituent elements or competing phases.However, the consideration of dynamical stability, ensur-∗ Xiao.Enda@nims.go.jp† Tadano.Terumasa@nims.go.jping that a compound does not undergo structural phasetransitions, is rarely incorporated into HTP frameworksdue to the computational cost and complexity of phononcalculations in magnetic systems.Since most of the intriguing properties of Heusler com-pounds are related to magnetism, it is vital to ensurethe thermal stability of their magnetic configurations atapplication temperatures. This stability is typically as-sessed using Tc. Previous studies have estimated Tc us-ing linear interpolation derived from experimental data[6, 14–16]. Some efforts have also employed mean-fieldtheory combined with exchange interaction constantsfrom first-principles calculations to estimate Tc [17, 18].However, the reliability and accuracy of Tc calculationshave not been thoroughly investigated against experi-mental results in a comprehensive Heusler space.The diversity of Heusler compounds arises not onlyfrom their composition but also from the possible struc-tures they can adopt for a given composition. Heuslercompounds with the X2Y Z composition can typicallycrystallize in either regular or inverse structures, assum-ing no chemical disorder. An empirical rule of structuralpreference in X2Y Z Heusler compounds, referred to asBurch’s rule in some studies, has been employed in var-ious works [18–22]. While this rule has slightly differentdescriptions in different works, the key point is that theinverse structure is preferred when the Y element is lo-cated to the right of the X element in the periodic table,and vice versa. This empirical rule has been validatedby HTP ab initio studies containing a few hundred com-2pounds [18, 22], but its validity for a broader composi-tional space is not clarified. Besides, Heusler compoundstypically crystallize in the cubic or tetragonal phase. Thetetragonal phase, also referred to as tetragonal distor-tion, is essential for a material to exhibit MCA, which isparticularly critical for developing materials with strongperpendicular magnetic anisotropy (PMA) [23].Recently, Heusler alloys have also emerged as promis-ing candidates for compensated ferrimagnet (CFiM) sys-tems. Compared to traditional ferromagnetic (FM) ma-terials, CFiM offers several advantages for spintronics ap-plications due to its low moment, such as faster switch-ing speeds, higher storage densities, and greater resis-tance to external magnetic fields. [24]. Notably, CFiMsystems have inequivalent magnetic sublattices, enablingconventional electrical reading and writing mechanismson FM systems, such as anomalous Hall effect (AHE),tunnel magnetoresistance (TMR), spin-transfer torque(STT), and spin-orbit torque (SOT) [24]. However, onlya few Mn-contained CFiM Heusler compounds have beensynthesized and investigated [25–34]. CFiM can be ob-tained from low-moment ferrimagnet (FiM) by adjustingthe chemical composition and concentration. Thus, alow magnetic moment FiM are potential candidates forCFiM, but only a few systems have been proposed byfirst-principles calculation [35–39]. Thus, an expandedlist of candidates taking stability into account is desired.In this work, we conducted a HTP ab initio searchfor stable Heusler compounds covering a broad range ofregular, inverse, and half structures in both cubic andtetragonal phases, significantly expanding the pool of ma-terials available for functional exploration. Stability wasassessed using conventional metrics such as formation en-ergy, hull distance, and magnetic transition temperature.Additionally, we incorporated phonon stability, a criti-cal factor omitted in previous high-throughput investiga-tions. To confirm the validity of our screening and mod-eling procedure, we benchmarked the proposed stabilitycriteria against a dataset of 189 experimentally synthe-sized compounds and the employed Tc calculation meth-ods against 59 experimental data points. This compre-hensive analysis identified 631 compounds that satisfy allstability criteria, marking them as promising candidatesfor further functional material exploration. Notably, weidentified 47 low-moment FiM systems that meet all sta-bility criteria. For these systems, we calculated the spinpolarization (SP), anomalous Hall conductivity (AHC),and anomalous Nernst conductivity (ANC), providing in-sights into their potential applications in spintronics andenergy harvesting devices.Our comprehensive HTP result also revealed signifi-cant linear relationships between Tc and magnetizationin 14 systems. We also observed correlations betweencompound stability and fundamental atomic propertiessuch as atomic radius and ionization energy. Addition-ally, we confirmed that inverse Heusler compounds gen-erally exhibit a lower electronegativity for the X elementcompared to the Y element, alongside a comparable co-valent radius difference between these elements. Finally,we confirmed that tetragonal distortion correlates with ahigh density of states at the Fermi level in the cubic phasefor X2Y Z compounds. We also found that this correla-tion extends to the studied half-Heusler compounds.II. METHODOLOGYA. Composition and structureThis HTP study investigates an extensive set of con-ventional Heusler compounds, as illustrated in Fig. 1 (a).The regular and inverse Heusler compounds share theX2Y Z composition, while half-Heusler compounds adoptthe XY Z composition. The regular and inverse struc-tures are identical when the X and Y elements are thesame, resulting in an X3Z composition. We consideredall combinations of elements where X and Y are tran-sition metals from the d-block (excluding Tc and Hg),and Z is a main group element from groups 13, 14, or15 in the p-block, as depicted in Fig. 1 (a). For the Xelement, we additionally considered La and Lu, whose 4forbitals are empty or fully occupied. This comprehen-sive screening resulted in a total of 27,864 compounds,including 9,072 regular, 9,072 inverse, 9,396 half-Heuslercompounds, and 324 X3Z compounds.For regular and inverse Heusler compounds, the cu-bic regular structure belongs to the Fm3m space group,with the X, Y , and Z atoms occupying the 8c ( 14 ,14 ,14 ),4b ( 12 ,12 ,12 ), and 4a (0, 0, 0) Wyckoff positions, respec-tively. The cubic inverse structure belongs to the Fm43mspace group, with the X1, X2, Y , and Z atoms at4c ( 14 ,14 ,14 ), 4b ( 12 ,12 ,12 ), 4d ( 34 ,34 ,34 ), and 4a (0, 0, 0)Wyckoff positions, respectively. Half-Heusler compoundswith XY Z composition adopt the cubic Fm43m spacegroup, where the X, Y , and Z atoms occupy the4c ( 14 ,14 ,14 ), 4b ( 12 ,12 ,12 ), and 4a (0, 0, 0) Wyckoff posi-tions, respectively. The tetragonal variants of regular, in-verse, and half-Heusler structures belong to the I4/mmm,I4m2, and I4m2 space groups for regular, inverse, andhalf structures, respectively.B. Workflow for high-throughput calculationThe workflow of the high-throughput ab initio calcula-tion is schematically shown in Fig. 1(b), which we elab-orate below.a. Cell parameter optimization The initial struc-tures of cubic Heuslers X2Y Z (or XY Z) were generatedusing a primitive cell containing one formula unit, wherethe initial lattice constant, aini, was set to the value re-ported in the OQMD for the same Heusler if available.When an OQMD calculated result was not available forX2Y Z (or XY Z), the corresponding aini value was es-timated by simply taking the average of the a values ofthe OQMD-calculated Heuslers X2Y Z ′ (or XY Z ′) that3X2YZ and XYZ Heuslers(a,b) Relax cell with various magneticconfigurations and tetragonal distortionsGenerate cubic structuresE < 0 &H < 0.3 eV/atom?UnstableNYUnstableN(c) Determine ground state and calculate E and H(d) Compute phonons within harmonic approximationPhonons stable?YIs magnetic?(e) Calculate Jij and Tc(e) Calculate spin polarization and anomalous Hall/Nernst conductivityTc > 300 K?Compensated ferrimagnet?YYNCandidates for future explorationNN27,8648,19119,6734,0114,180Regular X2YZ Inverse X2YZ Half XYZ(b)(a)2,655Y47Y27,864106,2351,356631725584FIG. 1. (a) The compositions covered in the high-throughput search and distribution of stable compositions. Under eachelement, the number of compounds that contain this element and meet the stability criteria (∆E < 0.0 eV/atom and ∆H <0.3 eV/atom, and ωmin = 0) is shown. (b) Workflow of the high-throughput search for stable Heusler compounds.share the same X and Y elements. The cell parameterswere then optimized by using the Vienna ab initio Simu-lation Package (VASP) [40–42]. For each composition, weconsidered various initial magnetic configurations follow-ing Ref. [43]. Specifically, for the regular Heusler X2Y Zand half Heusler XY Z, the X-site magnetic moment caneither be parallel or antiparallel to the Y -site magneticmoment. For each spin configuration, we tested two mag-nitudes for the local magnetic moment, i.e., |mi| = 1and 4 µB, to explore potential high-spin and low-spinstates. Hence, four initial magnetic configurations wereconsidered. In the case of the inverse Heusler X2Y Z, thespin moments at the two X sites can be antiparallel, sowe considered eight initial configurations in total. Afterthe structure optimizations were finished, the final ener-gies and magnetic moments were compared to identifyinequivalent (meta) stable states.b. Tetragonal distortion For each inequivalent(meta) stable magnetic state identified, the cubicprimitive cell was converted into a conventional cellcontaining two formula units, and the c-axis length waschanged from its original value by ±2%, ±10%, +30%,and +50%. The cell parameters were then optimizedfurther from these initial structures using VASP toexplore potential lower-energy tetragonal phases. Afterthe optimization, the structure with the lowest energywas identified as the ground state for the compositionX2Y Z (or XY Z).c. Thermodynamic stability For a compound to bestable against decomposition into its constituent ele-ments, the formation energy (∆E) must be negative.In addition, a compound is considered thermodynami-cally stable if its energy is lower than that of all possiblecompeting phases. This relative stability can be evalu-ated by the distance to the convex hull (∆H), and onlycompounds on the convex hull (∆H = 0) are thermo-dynamically stable in the strict sense. However, it ispossible that metastable phases (H > 0) at zero Kelvinbecome most stable at finite temperatures or under exter-nal strain. Indeed, previous studies reported that manymetastable compounds had been synthesized experimen-tally as long as H is reasonably small [44]. Hence, werelax the criterion and assume compounds to be ther-modynamically stable if ∆H < 0.3 eV/atom, which waschosen based on the analysis in the previous work. Thischoice is reasonable also for the Heuslers, as we willdemonstrate below. To summarize, compounds satisfy-ing both ∆E < 0.0 eV/atom and ∆H < 0.3 eV/atomwere deemed thermodynamically stable and subject todynamical stability analysis.d. Dynamical stability For assessing the dynamicalstability of Heuslers, phonon calculations were conductedwithin harmonic approximation using the ALAMODEpackage [45, 46]. The dynamical stability was assessedbased on the presence/absence of unstable phonon modeson the q points commensurate with the supercell size. Inthis work, a 2√2 × 2√2 × 2 conventional cell contain-ing 32 formula units was employed; hence, there are 32commensurate q points in the first Brillouin zone (BZ).If any of the eigenvalues of the dynamical matrix, i.e.,the squared phonon frequencies {ω2qν}, were negative atthe commensurate q points, the compound was assessedto be dynamically unstable. If all ω2qν values were non-negative, the compound was assumed to be dynamicallystable and subject to the subsequent property calcula-tions.e. Property calculations The dynamically stablecompounds were further split into magnetic and nonmag-netic systems. If the absolute sum of local magnetic mo-ment per formula unit,∑i |mi|, is larger than 0.1µB, the4system was considered magnetic. For the magnetic sys-tems, the Tc was evaluated using the spin-polarized rel-ativistic Korringa-Kohn-Rostoker (SPRKKR) code [47],and compounds having Tc higher than 300 K were identi-fied. Among them, we identified compounds that simul-taneously satisfy∑i |mi| > 0.5 µB and |∑i mi| < 0.5µB as CFiM candidates, for which the ANC and AHCwere computed systematically by using Wannier90 [48,49].C. Computational methodsDensity functional theory (DFT) calculations weremainly conducted using VASP [40, 50]. The projectoraugmented wave method and the generalized gradientapproximation (GGA) with the Perdew-Burke-Ernzerhof(PBE) functional were used [41, 42]. A plane-wave energycutoff of 520 eV was applied, and the BZ was sampledusing an automatically generated k-point mesh by set-ting KSPACING=0.2 Å−1 except for the body-centeredtetragonal (bct) lattice; for the bct lattice, the k-meshwas generated using the Python Materials Genomics (py-matgen) library with a reciprocal density of 450 Å3. TheMethfessel-Paxton smearing [51] with widths of 0.05 eVand 0.1 eV was employed for structural optimization andphonon calculations, respectively. For electron densityof states (DOS) calculations, the primitive cell was usedand k-mesh was generated using pymatgen with a denserdensity of 1000 Å3, and the tetrahedron method with theBlöchl corrections was used [52].The formation energy (∆E) was calculated as the en-ergy difference between the Heusler compound and itselemental constituents. The most stable elemental ref-erence structures were obtained from the OQMD andthen relaxed by VASP further using the DFT parametersdescribed above to obtain the energies of the elementalconstituents. The ∆H values were calculated using theformation energies computed for the Heuslers and thoseof the competing phases obtained from the OQMD (ver.1.6). The HTP optimization scripts utilized the pymat-gen package to generate input files for different initialstructures using recommended pseudopotentials [53, 54].Also, the Atomic Simulation Environment (ASE) toolkit[55] and the spglib [56] were employed for the structurefile manipulation and symmetry analysis, respectively.The phonon calculations were performed based on thesupercell approach, as implemented in ALAMODE. Eachatom in the supercell was displaced from its equilib-rium position by 0.01–0.02 Å, and the atomic forces inthe displaced configurations were computed using VASP.The second-order interatomic force constants (IFCs) be-tween the atoms in the supercell were fitted to thedisplacement-force dataset, and the dynamical matrixwas constructed from the IFCs to obtain the phonon fre-quencies. Phonon calculations for magnetic systems poseunique challenges, as small atomic displacements can al-ter the magnetic configuration in certain compounds. Tocircumvent this issue, we employed a two-step approach.First, a static DFT calculation was performed for thesupercell without displacing atoms, and the resultingcharge density was saved. Second, the DFT calculationsfor the displaced configurations were performed using thecharge density obtained in the first step as initial values.Here, we also fixed the total magnetic moment to thedesired value, as implemented by the NUPDOWN tagin VASP. We confirmed that this approach resulted inconsistent magnetic moments for the displaced configura-tions in almost all cases. In-house scripts were employedfor high-throughput phonon computations.The Tc values were determined within the mean-fieldapproximation using the exchange interaction constantsJij [57]:TC =23kBJmax, (1)where Jmax denotes the largest eigenvalue of the Jµν ma-trix defined as Jµν =∑j∈ν J0j . Here, the index 0 is afixed site in sublattice µ, and the summation runs over allsites in sublattice ν. Jij values were calculated using theLiechtenstein formula, implemented in SPRKKR [58].The exchange-correlation effects were treated within thePBE functional. The optimized structures by the VASPcalculations were employed, and the local magnetic mo-ments {mi} from the VASP calculations were used as theinitial values. We confirmed overall consistency betweenthe {mi} values obtained using SPRKKR and VASP, asshown in Fig. S2 in Supplemental Materials. The k-point mesh for the self-consistent and Jij calculationswas 28× 28× 28, and the basis set NL was set to 4. Thecutoff radius flag CLURAD was set to 5 in the Jij calcu-lation, with which we confirmed convergence in Tc. Re-sults employing the atomic sphere approximation (ASA)or full-potential (FP) method were computed [59]. TheHTP Tc calculation scripts utilized the ASE2SPRKKRpackage to generate input files from the optimized struc-tures [60, 61].The anomalous Hall conductivity (σxy) was calculatedusing the Kubo formula in terms of the Berry curvature,as implemented in the Wannier90:σxy = −e2h̄∫BZdk(2π)3∑nf (ϵnk) Ωn,xy(k), (2)where h̄, e, ϵnk, f , and Ωn,xy represent the reducedPlanck constant, positive elementary charge, eigenen-ergy, Fermi distribution, and Berry curvature, respec-tively. The Berry curvature is defined asΩn,xy(k) = −2h̄2 Im∑m(̸=n)⟨nk|v̂x|mk⟩⟨mk|v̂y|nk⟩(ϵnk − ϵmk)2 , (3)with v̂x (v̂y) being the kx (ky) component of the veloc-ity operator, and |nk⟩ representing the eigenstate. Fromthe DFT band structure, Wannier functions were gen-erated using the selected columns of the density matrix5(SCDM) method [62, 63]. The anomalous Nernst con-ductivity (αxy) at finite temperature T was calculatedfrom the energy-dependent anomalous Hall conductivityσxy(ϵ)[64, 65]:αxy(T ) =1eT∫dϵ (ϵ− µ)∂f(ϵ, T )∂ϵσxy(ϵ, T = 0) (4)where µ is chemical potential and set to Fermi energy inthe calculation. To realize integrations over the BZ inσxy calculation, a k-mesh of 150× 150× 150 points wasused. The αxy values were evaluated using σxy(ϵ) withina 0.8 eV energy window around the Fermi energy.III. RESULTSAfter relaxation, a total of 106,235 structures wereidentified, comprising 27,864 ground states and 78,371metastable states. The stability of the ground stateswas further evaluated. Among these ground states, 8,191(29.4%) satisfied the thermodynamic stability criteria of∆E < 0.0 eV/atom and ∆H < 0.3 eV/atom. Forthe 8,191 screened compounds, we attempted ab initiophonon calculations and successfully obtained phononfrequencies for 8,180 compounds. For the remaining 11cases, the phonon calculation results were not success-ful nor reliable either due to a convergence issue in DFTcalculations or a large fitting error in the force constantestimation. Hence, these compounds were excluded fromthe subsequent calculations. Out of the 8,180 phononcomputed compounds, 4,011 (14.4%) were identified dy-namically stable, which include 1,898 regular, 1,192 in-verse, 81 X3Z, and 840 half-Heusler compounds. Thesefindings underscore the importance of incorporating com-prehensive stability metrics for efficient material discov-ery, highlighting that phonon stability can significantlynarrow the candidates’ list. In Fig. 1(a), the number ofcompounds that contain the corresponding element andmeet the stability criteria is shown under each element.For magnetic applications, it is essential that com-pounds exhibit a magnetic configuration that is stable atoperational temperatures, typically room temperature.Among thermodynamically and dynamically stable com-pounds, 1,356 compounds have been identified as mag-netic satisfying∑i∈cell |mi| > 0.1 µB. Of these, 631 com-pounds have a Tc exceeding 300 K, including 240 regular,291 inverse, 24 X3Z, and 76 half-Heusler structures. TheFP Tc corrected by a factor of 0.85 were used. The vali-dation of Tc calculation method is included in subsectionIVA. The full list of stable magnetic compounds is in-cluded in the Supplemental Material.Among the stable compounds, 47 low-moment FiMsystems are identified, characterized by a total mag-netization |mtot| = |∑i∈cell mi| smaller than 0.5 µBand∑i∈cell |mi| larger than 0.5 µB . These systems arelisted in Table I. In previous work, it was proposedthat compounds containing 24/18 valence electrons forX2Y Z/XY Z composition and Mn element are promisingcandidates, and several such compounds were found byfirst-principle calculations [35–37]. Our finding expandsthe list of low-moment Mn-containing compounds. Wealso find that Cr-containing compounds Cr2YZ (Y=Pd,Pt, Rh) can also exhibit low moments. Two V-containingcompounds and a Ti-containing compound are also iden-tified. It should be noted that the Slater-Pauling rule isnot valid for some compounds, while the valence electronnumber is close to the magic number 24 or 18. As sug-gested by the appearance of MnCrAs compound, we ex-pect that quaternary type Heusler compounds containingMn and Cr can exhibit small |mtot| values, potentiallyproviding more CFiM candidates.To aid in the discovery of functional materials, we cal-culated the SP, AHC, and ANC to evaluate their poten-tial performance, as listed in Table I. Notably, 17 com-pounds exhibit a SP greater than 0.6, which correspondsto a TMR of 112% according to the Jullière’s formula [66].Among these, Cr2MnAs, MnCrAs, Mn2OsGe, Ti2CrSn,and Cr2IrAl stand out with the SP values of 0.85, 0.87,0.89, 0.96, and 0.96, respectively. In a recent work,low-moment FiM compounds Ti2MnZ (Z=Al, Ga) werereported to exhibit promising AHC (253/268 S cm−1)and ANC at room temperature (1.31/0.94 Am−1 K−1)by first-principle calculation [38, 39]. Among the newlyidentified compounds, 12 compounds exhibit an AHC ex-ceeding 250 S cm−1 and 8 compounds exhibit an ANC ex-ceeding 1Am−1K−1. Given that the Fermi level can shiftwith variations in chemical composition and concentra-tion in tuning for CFiM, we also present the maximumAHC and ANC values within an energy window of 250meV around the Fermi level, as detailed in Table I.As an example of the identified functional CFiM com-pounds, we show the calculated electronic band struc-tures with and without spin-orbit coupling, the energy-dependent AHC/ANC, and the phonon band structureof inverse Mn2RhAl in Fig. 2. The AHC shows a smallvariance around the Fermi level, suggesting the potentialthat total magnetization can be tuned by doping with-out lowering its AHC. The structure is tetragonally dis-torted with a = 5.43 Å and c = 7.30 Å whose localmagnetic moments are shown in Table I. This structureexhibits ∆E = −0.48 eV/atom and ∆H = 0.07 eV/atom,indicating its potential synthesizability. The energy ofthe structure is 0.18 eV/atom lower than the most sta-ble regular structure having the same chemical formulaMn2RhAl. Also, the ∆H value is lower than those ofthe competing half Heuslers MnRhAl and RhMnAl by0.60 and 0.43 eV/atom, respectively, suggesting the pre-ferred stability in the inverse structure. We also foundanother metastable inverse Mn2RhAl that is less sta-ble than the identified CFiM candidate only by 0.006eV/atom. This competing structure is slightly distortedwith a = 5.92 Å and c = 5.94 Å, and its total mag-netic moment is 1.90µB/cell, which are consistent withthe previously reported theoretical value for the cubicinverse Mn2RhAl [67]. Since the energy difference be-tween the two inverse structures is small, experimental6TABLE I. List of stable ferrimagnets with a total magnetization smaller than 0.5 µB/f.u. and an absolute sum of localmagnetic moment larger than 0.5 µB/f.u. The table provides the formation energy ∆E (eV/f.u.), distance to the convex hull∆H (eV/f.u.), total magnetic moment mtot (µB/f.u.), local magnetic moments mloc (µB), and magnetic critical temperatureTc (K) computed using the full-potential approach. Additionally, it includes the spin polarization SP, and electron density ofstates DOS (states eV−1f.u.−1) at the Fermi level, anomalous Hall conductivity σxy (S/cm), anomalous Nernst conductivity αxy(A/Km), and the maximum anomalous Hall/Nernst conductivity σmaxxy /αmaxxy within an energy window of 250 meV around theFermi level. In type column, r/i/h represents regular/inverse/half, and c/t represents cubic/tetragonal. For easier comparison,the total magnetic moment in units of emu/cm3 and emu/g are also included in the full list of statble compouds in SupplementalMaterial.composition type ∆E ∆H mtot mloc Tc SP DOS σxy σmaxxy αxy αmaxxyCr2AuAl it −0.04 0.16 0.06 [−2.90, 2.99, 0.01,−0.01] 1932 0.52 3.28 −658.02 −718.50 0.42 −1.78Cr2IrAl ic −0.28 0.19 0.01 [−2.05, 2.11, 0.00, 0.00] 1310 0.96 6.10 −325.17 −603.50 −0.11 2.37Cr2IrGe it −0.10 0.11 0.29 [2.53,−2.38, 0.12, 0.00] 1292 0.16 6.08 14.02 663.00 1.51 2.06Cr2MnAs rt −0.03 0.10 0.03 [−1.42,−1.42, 2.83, 0.08] 760 0.85 4.81 342.97 689.58 0.83 −2.30Cr2NiGe it −0.05 0.14 0.01 [2.29,−2.30, 0.00, 0.00] 1557 0.81 5.14 52.53 199.49 −0.12 1.00Cr2PdAl it −0.16 0.29 0.10 [2.88,−2.83,−0.01, 0.01] 1802 0.26 4.89 129.35 385.71 0.13 −1.19Cr2PdAs it −0.03 0.17 0.21 [3.04,−2.92, 0.05, 0.02] 1518 0.02 3.91 −86.87 −390.94 0.42 −1.35Cr2PdGa it −0.11 0.22 0.10 [2.93,−2.87, 0.00, 0.00] 1813 0.12 5.02 65.30 66.14 0.21 −0.51Cr2PdGe it −0.08 0.19 0.04 [−2.80, 2.85, 0.00, 0.00] 1811 0.68 3.34 −55.15 −66.43 0.09 −0.58Cr2PdSn it −0.02 0.27 0.08 [−3.07, 3.15, 0.01, 0.00] 1759 0.57 2.94 −18.76 −69.87 0.19 0.43Cr2PtAl it −0.32 0.20 0.15 [2.78,−2.71, 0.01, 0.00] 1701 0.31 4.91 235.79 246.94 0.09 −1.04Cr2PtGa it −0.23 0.10 0.11 [2.84,−2.78, 0.02, 0.00] 1734 0.19 4.75 100.26 −266.89 0.09 −1.26Cr2PtGe it −0.16 0.08 0.05 [−2.72, 2.80,−0.03, 0.00] 1727 0.63 3.34 −149.56 −241.16 −0.57 1.17Cr2PtIn it −0.05 0.21 0.00 [3.15,−3.16,−0.01, 0.00] 1660 0.23 4.81 −47.10 176.52 0.11 −1.01Cr2PtSi it −0.27 0.18 0.01 [2.46,−2.52, 0.04, 0.00] 1633 0.68 4.02 58.57 321.61 0.34 1.08Cr2PtSn it −0.10 0.21 0.04 [−3.01, 3.07,−0.02, 0.00] 1754 0.61 2.83 −181.73 −186.59 0.12 0.45Cr2RhGe it −0.14 0.14 0.41 [2.66,−2.42, 0.13, 0.00] 1482 0.12 5.82 −280.10 −315.97 −1.20 −1.81Cr2RhSb it −0.04 0.21 0.03 [−2.83, 2.92,−0.06, 0.01] 1621 0.67 3.52 9.60 −136.89 0.14 1.59Cr2RhSn it −0.04 0.23 0.11 [2.95,−2.92, 0.05, 0.00] 1544 0.32 4.91 −185.02 −412.51 0.01 2.03CrMnAs hc −0.04 0.14 0.00 [−2.50, 2.60, 0.00] 2244 0.61 2.82 5.41 202.19 0.32 0.89Mn2AgAl ic −0.02 0.17 0.33 [−3.08, 3.38, 0.03, 0.01] 1218 0.56 4.82 220.09 297.21 0.12 −1.22Mn2AuIn it −0.00 0.12 0.03 [3.61,−3.60, 0.00, 0.01] 782 0.11 5.45 −25.94 531.99 1.23 1.48Mn2CuAl it −0.18 0.01 0.20 [−2.56, 2.77, 0.00,−0.01] 1264 0.60 4.72 11.41 56.12 0.15 0.16Mn2CuGa it −0.11 0.00 0.35 [−2.75, 3.06, 0.02, 0.00] 1205 0.57 4.70 11.09 301.80 0.13 1.03Mn2Ge ht −0.04 0.09 0.00 [−2.27, 2.36,−0.04] 1751 0.69 3.28 9.30 119.51 −0.20 −0.36Mn2IrAl it −0.51 0.00 0.22 [−2.81, 3.04,−0.01, 0.00] 947 0.38 5.27 −846.32 −969.32 −0.91 −1.44Mn2IrGa it −0.39 0.00 0.20 [−2.93, 3.13,−0.02, 0.01] 984 0.48 5.12 −836.28 −881.77 −0.92 −1.86Mn2IrIn it −0.15 0.01 0.05 [−3.32, 3.38,−0.02, 0.01] 888 0.64 4.95 −686.49 −717.11 0.11 0.94Mn2NiGe it −0.22 0.00 0.37 [−2.46, 2.75, 0.06, 0.00] 917 0.32 5.02 −14.88 −97.78 −0.14 −0.69Mn2NiSn it −0.08 0.07 0.38 [−3.00, 3.31, 0.04, 0.01] 761 0.40 6.08 167.06 273.11 0.38 0.81Mn2OsGe it −0.13 0.00 0.03 [2.74,−2.85, 0.16,−0.01] 706 0.89 3.54 194.50 497.76 0.92 −1.77Mn2PdIn it −0.15 0.11 0.28 [−3.46, 3.67, 0.04, 0.01] 346 0.21 5.84 33.27 177.18 −0.01 1.53Mn2PdSn it −0.19 0.10 0.24 [−3.37, 3.57, 0.02, 0.01] 743 0.13 4.80 22.92 −141.62 0.18 −0.52Mn2PtSn it −0.23 0.09 0.04 [3.48,−3.43, 0.00, 0.00] 724 0.11 4.16 153.43 250.79 0.27 1.28Mn2ReGe it −0.07 0.05 0.22 [2.48,−2.53, 0.27,−0.01] 681 0.31 5.66 −564.75 −626.18 1.56 −1.76Mn2RhAl it −0.48 0.07 0.23 [−2.90, 3.11, 0.01, 0.00] 1066 0.50 5.64 −510.42 −658.36 −0.13 1.54Mn2RhGa it −0.40 0.00 0.16 [−3.06, 3.21, 0.00, 0.01] 1102 0.56 5.52 −221.33 −548.43 0.33 1.12Mn2RhIn it −0.19 0.03 0.01 [−3.46, 3.46, 0.00, 0.01] 996 0.73 4.81 71.17 518.14 −0.60 −1.99Mn2RuGa it −0.25 0.00 0.23 [3.02,−3.01, 0.23,−0.02] 1170 0.00 4.69 −191.07 377.05 −1.17 −1.59Mn2RuIn it −0.00 0.05 0.14 [3.32,−3.35, 0.18,−0.02] 1147 0.13 4.99 −97.71 −625.15 −1.08 −1.78Mn2RuSb it −0.07 0.03 0.20 [−2.94, 3.09, 0.01, 0.01] 568 0.41 6.35 −357.75 −692.35 −0.30 1.14Mn2RuSn it −0.11 0.00 0.02 [3.13,−3.22, 0.13,−0.01] 850 0.80 5.01 278.54 579.82 −1.72 2.02Mn2Si hc −0.19 0.18 0.01 [1.29,−1.33, 0.02] 1137 0.67 0.53 −0.04 −304.74 0.05 −0.95MnCrAs hc −0.13 0.05 0.00 [−1.45, 1.45,−0.02] 1690 0.87 2.82 125.80 448.74 −0.16 −1.05Ti2CrSn ic −0.13 0.19 0.00 [1.33, 0.98,−2.64, 0.02] 1691 0.96 3.73 56.23 85.79 −0.09 0.33V2ScGa ic −0.05 0.25 0.30 [−1.43, 1.57, 0.15, 0.00] 585 0.47 9.21 −468.32 −468.32 −1.08 −1.98V2TiSn ic −0.08 0.14 0.04 [1.42,−0.67,−0.69, 0.00] 700 0.22 1.74 −25.56 88.31 0.07 −0.357Γ X Y Σ Γ Z Σ1 N P Y1 ZWave Vector−0.4−0.20.00.20.4E−EF(eV)Spin upSpin dnSOC−500 0 500AHC (S/cm)−1 0 1ANC (A/K/m)Γ X Y Σ Γ Z Σ1 N P Y1 ZWave Vector050100150200250300350ω(cm−1 )FIG. 2. Calculated electronic band structures with and without spin-orbit coupling, AHC, ANC, and the phonon band structureof inverse tetragonal Mn2RhAl.8Mn2VZAlGaInTlSiGeSnPbP AsSbBiMn2CoZAlGaInTlSiGeSnPbP AsSbBiTi2FeZAlGaInTlSiGeSnPbP AsSbBiV2FeZAlGaInTlSiGeSnPbP AsSbBi−0.10 −0.05 0.00 0.05 0.10All ground states(eV/atom)FIG. 3. Structural preference and stability of Mn2VZ,Mn2CoZ, Ti2FeZ and V2FeZ compounds. The pie chartslices are arranged by the Z element group. The inner cir-cle indicates the stability categorized into three levels, whilethe outer-circle color map represents the energy differenceEreg − Einv in the range [−0.1 , 0.1 ] eV/atom. Blue and redindicate preference for regular and inverse structures, respec-tively. Values outside this range are capped at the boundaryvalues.realization of the CFiM candidate would require carefuloptimization of the synthesis conditions.Besides the discovery of promising candidates, thecomprehensive HTP dataset also provides insights intothe stability and structural preference in Heusler space.For example, X2Y Z has a preference for either the regu-lar structure or the inverse structure over the other. Forvisualizing the distribution of regular and inverse struc-tures and their relative stability, pie charts were used, asshown in Fig. 3, using Mn2VZ, Mn2CoZ, Ti2FeZ, andV2FeZ as examples. The pie chart slices are arranged bythe Z element. The inner circle indicates stability, whilethe outer-circle color map represents the energy differ-ence Ereg − Einv. In the inner circle, the stability levelis classified into three categories based on ∆E, ∆H, andωmin. Here, ωmin is the minimum value of the computedphonon frequencies on the commensurate q points and isshown as a negative value when it is imaginary. Hence,ωmin = 0 holds when the compound is dynamically sta-ble. Given that the preference for regular or inverse struc-tures is significant when the energy difference exceeds0.1 eV/atom, the color map is capped at 0.1 eV/atom.In these examples, Mn2VAl and Mn2VGa are both sta-ble and prefer the regular structure, while Mn2CoAl,Mn2CoGa, Mn2CoGe, Mn2CoSn, and Mn2CoSb are sta-ble and exhibit a preference for the inverse structure,consistent with experimental data [6]. The coexistence ofregular and inverse structures has also been experimen-tally observed in compounds such as Ti2FeAl, Ti2FeGa,V2FeAl, and V2FeGa [20, 21]. The pie charts in Fig. 3show the small energy differences (Ereg − Einv) in thesecases, which explain the coexistence of regular and in-verse structures.Using this visualization style, we generated a compre-hensive stability and regular/inverse preference map forallX2Y Z compounds, shown in Fig. 4. The x- and y-axescorrespond to the X and Y elements, respectively, whileeach pie chart represents the stability and structural pref-erences of 12 compounds with varying Z elements. Thisvisualization style provides a clear and comprehensiveoverview of stability and structural preferences over X,Y , and Z compositions. Most stable inverse compoundsare distributed in the lower left corner of the map, align-ing with the empirical rule. However, a significant num-ber of stable regular compounds are also located in thisregion. A detailed discussion of the regular/inverse pref-erence is provided in subsection IVD.We also generated maps for other competition pairs.For compounds with the same X, Y , and Z elements,the preference between X2Y Z and XY Z compositionsis determined by the Hull distance difference betweenthe half-Heusler and the lower energy regular or inverseHeusler compound. For a given composition and atomicarrangement, the competition between cubic and tetrag-onal phases is determined by the energy difference be-tween the two phases. Comprehensive maps illustrat-ing the competition between X2Y Z/XY Z compositionsand cubic/tetragonal phases are provided in the Supple-mental Material. These visualizations offer a clear andcomprehensive overview to aid further study in Heuslercompounds space.IV. DISCUSSIONA. Magnetic critical temperature (Tc)The Tc values were computed within the mean-field ap-proximation where the exchange coupling constants (Jij)were obtained using the magnetic force theorem, as im-plemented in the SPRKKR code. The DFT calculationsbased on the KKR method were performed either withinthe ASA or using the FP method. To assess the reliabilityof the computed Tc values, we first compare the resultswith experimental data from previous high-throughputstudies by Sanvito et al. [6] and Hu et al. [18], as shownin Fig. 5 (a). Overall, the calculated Tc values show goodagreement with the experimental data. The Tc values ob-tained using the ASA exhibit considerable scatter, some-times exceeding and sometimes falling below the exper-imental values. In contrast, the FP-base Tc values areconsistently higher than the experimental data, whichis reasonable given the mean-field approximation’s ten-dency to overestimate Tc [68]. 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AsSbBiPdAlGaInTlSiGeSnPbP AsSbBiPtAlGaInTlSiGeSnPbP AsSbBiCuAlGaInTlSiGeSnPbP AsSbBiAgAlGaInTlSiGeSnPbP AsSbBiAuAlGaInTlSiGeSnPbP AsSbBiZnAlGaInTlSiGeSnPbP AsSbBiCdAlGaInTlSiGeSnPbP AsSbBielement XelementYFIG. 4. Comprehensive stability and structural preference map for X2Y Z Heusler compounds. The x and y axes represent theX and Y elements, respectively. The order of elements is sorted by the group number for convenient comparison with empiricalBurch’s rule. Each pie chart corresponds to 12 compounds with the same X and Y elements but different Z elements arrangedin the same style as Fig. 3. The inner circle indicates stability, while the outer circle represents Ereg −Einv in the range [−0.1 ,0.1 ] eV/atom. Blue/red indicates a preference for regular/inverse structures. Values outside this range are capped.factor 0.85 to the FP-based Tc values, the results alignwell with the experimental data, achieving an R2 scoreof 0.87. This adjustment compensates for the overesti-mation inherent in mean-field theory, and the result issatisfactory for high-throughput screening purposes.Additionally, a consistency check of the calculationmethod was performed by comparing our calculated Tcvalues with those reported by Hu et al. in Ref. [18],which also utilized the SPRKKR code and shown asa black empty circle in Fig. 5. In Ref. [18], Tc val-ues were primarily calculated using the ASA, whereasthe FP-based computation was conducted only when themagnetization from the ASA results was inconsistentwith the VASP magnetization. Consequently, some Tcvalues align with our ASA results, while others alignwith our FP-based calculations. Additionally, thereare four cases—Co2MnAl, Co2MnGa, Mn2CoGa, andMn2CoSn—where the results obtained by Hu et al. dif-fer from both the ASA and FP values in our study.These discrepancies arise from differences in the iden-tified ground states.A previous study identified a linear relationship be-tween the experimental Tc and total magnetization inCo2-based ferromagnetic Heusler compounds across 10100 250 500 750 1000 1250 1500Experimental Tc (K)0250500750100012501500CalculatedT c(K)(a) KKR-ASAKKR-FPHu et al .0.85×KKR-FP0 2 4 6∑i |mi |0200400600800100012001400CalculatedT c(K)(b) Co2YZSc2YZV2YZX2HfZXNbZXTaZFIG. 5. (a) Comparison of calculated and experimental Tc.Calculated Tc values using ASA and FP methods are shown,along with FP results corrected by a factor of 0.85. Exper-imental data are from Refs. [6, 18]. (b) The distribution ofcalculated Tc via FP approach as a function of absolute sumof local magnetic moment∑i |mi| in 6 Heusler compoundsystems.different compositions [15]. Our calculated Tc values,obtained via the FP approach, exhibit the same trendin certain systems. Fig. 5 (b) illustrates the distribu-tion of Tc values for six Heusler compound systems asa function of the absolute sum of local magnetic mo-ments,∑i |mi|. The total magnetization is generalizedto∑i |mi| since there are FiM and AFM compounds inour HTP results. Besides Co2Y Z compounds, this linearrelationship is also observed in other Heusler systems, in-cluding Sc2Y Z, V2Y Z, X2HfZ, XNbZ, and XTaZ, asshown in Fig. 5(b). Notably, while the linear trend holdsacross these systems, the slope varies. For instance, theslope of XNbZ compounds is significantly steeper thanTABLE II. Linear fit coefficients and R2 scores for the cor-relation between Tc and the absolute sum of local magneticmoments,∑i |mi|. The linear relationship is expressed asTc= a∑i |mi| + b. Results are presented for cases with datapoints number N larger than 10 and R2 scores exceeding 0.7.system N a b R2 system N a b R2X2CoZ 96 217 −145 0.77 Lu2Y Z 33 146 −84 0.85Sc2Y Z 79 168 −36 0.87 Y2Y Z 31 116 −25 0.85Ti2Y Z 64 363 −163 0.71 X2HfZ 28 155 27 0.78V2Y Z 64 241 −127 0.72 X2NbZ 24 164 8 0.79X2TiZ 60 164 −41 0.73 Co2Y Z 132 220 −24 0.83CoY Z 39 252 −69 0.77 Fe2Y Z 125 211 −141 0.73X2ScZ 37 124 38 0.72 XNbZ 12 363 −34 0.80that of Sc2Y Z compounds. Overall, we identified five sys-tems with an R2 score exceeding 0.8 and nine additionalsystems with an R2 score above 0.7. The correspondinglinear fit coefficients and R2 scores are listed in Table II.For systems with high R2 values, this linear relationshipcan be used to estimate the Tc of new compounds. Thedistributions of Tc values versus∑i |mi|, categorized byX or Y element species, are provided in Fig. S3 and S4 inthe Supplemental Material, along with the correspondinglinear fit coefficients and R2 scores.The distribution of ferrimagnet (FiM) compoundsamong the 1,356 magnetic materials is noteworthy.Specifically, 12 (1%) are antiferromagnetic, and 636(47%) are FiM. All FiM compounds contain one or moreof the following elements: Mn, Cr, Fe, V, Co, or Ti. Thecounts of FiM compounds containing Mn, Cr, Fe, V, Co,and Ti are 262, 123, 122, 120, 92, and 55, respectively.In comparison, the counts of ferromagnetic compoundscontaining these elements are 140, 41, 163, 47, 170, and95, respectively. The ratio of FiM ordering is high incompounds containing Mn, Cr, and V. This is in agree-ment with a synthesis of such compounds in experiments[69–71]. Detailed distributions of magnetic ordering, cat-egorized by element species and full/inverse/half Heuslerstructures, are presented in Fig. S5 of the SupplementalMaterial.B. Refine stability criteria using experimental dataThe stability of Heusler compounds was evaluated us-ing three criteria: formation energy (∆E), Hull distance(∆H), and minimum phonon frequency (ωmin). The dis-tribution of X2Y Z composition compounds across thesemetrics is shown in Fig. 6. For each X2Y Z composition,the structure (regular or inverse) with lower energy is se-lected. Formation energy values are evenly distributed,peaking around 0 eV/atom. For minimum phonon fre-quency, a sharp peak at 0 cm−1 reflects that approxi-mately half of the thermodynamically stable compoundsare also phonon-stable.To assess the reliability of these stability criteria,we compare the stable compound dataset from first-11−1.0 −0.5 0.0 0.5 1.0Formation energy (eV/atom)0200400600800CountCompounds not in ICSDCompounds in ICSD0.0 0.2 0.4 0.6 0.8 1.0 1.2Hull distance (eV/atom)0100200300400Count−400 −350 −300 −250 −200 −150 −100 −50 0minimum phonon frequency (cm−1)0500100015002000Count−1.0 −0.5 0.00200.0 0.1 0.2 0.30255075100−400 −300 −200 −100 0050100FIG. 6. Distribution of X2Y Z compounds ground states overformation energy, Hull distance, and minimum phonon fre-quency. Minimum phonon frequency distribution includesonly compounds with ∆E < 0.0 eV/atom and ∆H <0.3 eV/atom. The number of compounds included in ICSDdatabase are shown as yellow bins.020406080100Percentage (%) in HTP100110120130140150160countsinICSD∆E < 0.2∆E < 0.0 ∆H < 0.3∆H < 0.15ωmin > −30ωmin = 0∆H < 0.22∆H < 0.15∆H < 0.1∆E < 0.0ωmin > −30ωmin = 0no correlation02000400060008000counts in HTP60708090100Percentage(%)inICSDFIG. 7. Performance of ab initio stability criteria validatedagainst ICSD experimental data. The x axis shows the countsand percentages of HTP compounds satisfying sequentiallyapplied stability criteria, while the y axis indicates the countsand percentages of ICSD compounds included in the subset.If a criterion has no correlation with stability, it would fall onthe diagonal line colored as gray.principles predictions to experimentally synthesized com-pounds from the Inorganic Crystal Structure Database(ICSD) [72]. The ICSD contains 169 X2Y Z and 21 XY ZHeusler compounds. The distributions of ∆E, ∆H, andωmin of the ICSD-registered X2Y Z compound are over-laid in Fig. 6. Additional figures for XY Z compoundsare provided in the Supplemental Material.Among the 169 X2Y Z compounds from the ICSD, 159(94%) exhibit formation energies below 0 eV/atom, un-derscoring that negative formation energy is a robust nec-essary condition for stability prediction with large recall.A few ICSD compounds have positive calculated forma-tion energies of up to 0.2 eV/atom. Since formation en-ergy is calculated at 0K, the observed stability of thesecompounds in experiments might be due to the stabiliz-ing effect of entropic contributions at finite temperatures.Additionally, the positive values could be attributed touncertainties in DFT calculations. In a compound searchtask prioritizing the identification of as many candidatesas possible, a relaxed threshold of ∆E is a reasonablechoice.The hull distance values for most ICSD-registeredHeuslers are close to zero. A few ICSD data extendup to ∆H = 0.22 eV/atom, which is lower than the∆H < 0.3 eV/atom threshold used in the HTP workflow.Using a threshold of ∆H < 0.22 eV/atom enhances pre-cision without compromising recall of ICSD data, whilea stricter threshold of ∆H < 0.10 eV/atom can narrowdown the candidate list by 62% compared to ∆H < 0.22eV/atom list, at the cost of a smaller recall (89%).The distribution of ωmin for compounds with ∆E < 0eV/atom and ∆H < 0.3 eV/atom is presented in Fig. 6,including 159 compounds from the ICSD. Imaginaryphonon modes (represented as ωmin < 0) indicate struc-tural phase transitions and thus render the structuresdynamically unstable, as observed in compounds suchas Pd2TiSn, Pd2ZrSn, and Pd2HfSn at low tempera-tures [73]. Among the ICSD compounds, 124 (78%) ex-hibit ωmin = 0, affirming that the calculated phonon sta-bility serves as a good criterion for stability. The remain-ing 35 ICSD compounds display ωmin values ranging from−0.5 to −71.4 cm−1 and two points at −111.0 cm−1 and−308.9 cm−1. The number of compounds with negativeωmin values decreases as the magnitude of ωmin becomeslarger.These predictions are based on phonon dispersionwithin the harmonic approximation. It should be notedthat this approach does not account for anharmonic ef-fects, which can stabilize systems with imaginary phononmodes at finite temperatures [45, 73, 74]. The 35 ICSDcompounds with negative ωmin values may become dy-namically stable when anharmonic effects are includedat finite temperatures. However, due to the complex-ity and high computational cost associated with treat-ing anharmonic effects, a more practical strategy is touse ωmin computed at the harmonic level and choose athreshold value at a balance of precision and recall basedon the specific application. For instance, a threshold of12ωmin = 0 cm−1 is recommended for tasks prioritizing re-liable candidates, whereas a threshold of ωmin > −70cm−1 is suitable for identifying a broader range of poten-tial candidates.As shown in Fig. 6, applying stricter criteria increasesthe precision of stability predictions, as the ratio of re-moved compounds is larger in the HTP than the ICSDdata, while recall is reduced. Figure 7 illustrates thenumber of the HTP compounds that meet various sta-bility criteria applied sequentially and the number ofthe ICSD-registered Heuslers included within this sub-set. The x-axis represents the numbers/percentages ofHTP compounds that satisfy various stability criteriaapplied sequentially. The number decreases as the cri-teria become stricter. The y-axis represents the num-bers/percentages of ICSD compounds included in eachset of HTP compounds. The location of the points in theplot indicates the trade-off between precision and recall.If a criterion has no correlation with stability, it wouldfall on the gray diagonal line. By switching the order inwhich ∆E and ∆H are applied, it is clear that ∆H isa more effective criterion for identifying thermodynami-cally stable compounds.When using the criteria ∆E < 0.0 eV/atom and∆H < 0.3 eV/atom, 4,058 (43%) of HTP compounds arepredicted to be stable, and 159 (94%) ICSD compoundsare correctly identified as stable. When ωmin = 0 cm−1 isadded as a criterion, 2,173 (23%) of HTP compounds arepredicted to be stable, and 124 (78%) ICSD compoundsare correctly identified as stable. This demonstrates thevalidity of the stability criterion choice in our workflow.From Fig. 7, we can identify optimal criteria for differ-ent HTP screening task types quantitatively. In a screen-ing task prioritizing the identification of as many stablecompounds as possible, a relaxed threshold of ∆E is areasonable choice. Using criteria ∆E < 0.2 eV/atom and∆H < 0.22 eV/atom, 3,968 (42%) of HTP compoundsare predicted to be stable, and all ICSD compounds arecorrectly identified as stable. This set of criteria offersthe best recall. Conversely, with the criteria ∆E < 0.0eV/atom, ∆H < 0.10 eV/atom, and ωmin = 0 cm−1,1,057 (10%) HTP compounds are predicted to be stable,and 118 (70%) ICSD compounds are correctly identifiedas stable. This set of criteria is expected to provide goodprecision, albeit with a trade-off in recall.Similar trends are observed for half-Heusler com-pounds, where 1181 (13%) HTP compounds meet the cri-teria of ∆E < 0.0 eV/atom and ∆H < 0.3 eV/atom. Of21 half-type ICSD compounds, 19 meet the same criteria.The exception CuMnSb has small ∆E = 0.05 eV/atomand ∆H = 0.07 eV/atom. While the other exceptionCoCrAl has large ∆H = 0.44 eV/atom, the reported ex-perimental structure of CoCrAl has Cr occupying 50%( 12 ,12 ,12 ) and 50% ( 34 ,34 ,34 ) Wyckoff positions, which isnot standard half Heusler structure used in our compu-tation [75]. The stability observed in experiments maybe due to the disorder. When ωmin = 0 cm−1 is added asa criterion, 842 (9%) HTP compounds are predicted tobe stable, and 18 (85%) ICSD compounds are correctlyidentified as stable.C. Stability and atomic featureFrom the HTP calculation results, a correlation be-tween stability and atomic properties is identified. ForX2Y Z compounds, stability is favored when the Z ele-ment has a small atomic radius and low ionization en-ergy. Figure 8 shows the distribution of X2Y Z typecompounds over Z element species, along with the Zelement’s first ionization energy (I1) and atomic radius(ratom) [76–78]. Each bin represents the counts of com-pounds that contain the corresponding Z element. Thecompounds are divided into different stability groups in-dicated by various colors. The local and global trendsin stable compound distribution are consistent with thetrends in I1 and ratom, respectively. For comparison,the distribution of half-Heusler compounds is also shown,where no such correlation is found. Among the phonon-stable X2Y Z compounds, 51% contain either Al, Ga, orIn as the Z element, and 26% contain either Si, Ge, orSn. These elements should be prioritized for discoveringstable functional Heuslers. This observation is expectedto be valid for more general Heusler compounds, such asquaternary and off-stoichiometric types.0250500750X2YZcountAll ground states ∆E < 0.0; ∆H < 0.3 ∆E < 0.0; ∆H < 0.3; ωmin = 00250500750XYZcount7.510.0I 1(eV)Al Si P Ga Ge As In Sn Sb Tl Pb Bi100150r atom(Å)FIG. 8. Distributions of X2Y Z and half-type compounds overZ element species, with different stability groups shown invarious colors. First ionization energy (I1) and atomic radius(ratom) of Z elements are sourced from [76–78].In half-Heusler compounds, stability is higher when theX element has a smaller atomic radius compared to the Yelement. Figure 9 shows the distribution of half-Heuslercompounds over X and Y element species, along withthe atomic radii of the X and Y elements. It is evidentthat stable compounds are concentrated in the regionwhere the X element has a smaller atomic radius, and130200X2YZtypeXelementAll ground states ∆E < 0.0; ∆H < 0.3 ∆E < 0.0; ∆H < 0.3; ωmin = 00200X2YZtypeYelement0200HalftypeXelement0200HalftypeYelementSc Ti V Cr MnFe Co Ni Cu Zn Y Zr NbMoRuRhPd AgCd La Lu Hf Ta W ReOs Ir Pt Au Hg1.501.75r atom(Å)FIG. 9. Distributions of X2Y Z and half-type compounds overX or Y element species, with different stability groups shownin various colors. Atomic radius (ratom) of X and Y elementsare sourced from [76, 77].the Y element has a larger radius. For comparison, thedistribution of X2Y Z compounds is also shown in Fig-ure 9, where no such correlation is observed. Notably,94% of stable half-Heusler compounds satisfy the con-dition rXatom ≤ rYatom. This provides a robust necessarycondition for selecting stable half-Heusler candidates.D. Regular or inverse structure preference inX2Y Z compoundUnderstanding the preference for regular or inversestructures in X2Y Z compounds is crucial for tailoringtheir functional properties. Figure 10(a) shows the dis-tribution ofX2Y Z compounds based on the energy differ-ence between regular and inverse structures (Ereg−Einv).Overall, the regular structure is generally preferred acrossall X2Y Z compounds analyzed. This trend remainedconsistent when various stability criteria were applied.The structural preference analysis for regular or inverseconfigurations in X2Y Z compounds is often guided byBurch’s rule [18–22] According to this empirical rule, ifthe Y element is positioned to the left of theX element inthe periodic table, the compound is more likely to adopta regular structure. Conversely, if the Y element is to theright of the X element, an inverse structure is favored.This trend can be quantitatively represented using elec-tronegativity differences (χX − χY ), as electronegativitygenerally increases from left to right across the periodictable [18]. A negative value of χX −χY indicates that Yis to the right of X, favoring an inverse structure.This relationship is examined using our HTP calcula-tion results and illustrated in Fig. 10(b). The preferenceis shown by color-coded energy difference (Ereg − Einv),along with the electronegativity difference (χX−χY ) andcovalent radius difference (rXcov − rYcov) between X and Yelements are shown on the x- and y-axes, respectively.The results are shown for all compounds (5,388) and forthe subsets (1,294) meeting the stability criteria. Inversestructures tend to occur when χX − χY < 0, in accor-dance with Burch’s rule. This pattern remains even afterconsidering the stability criteria.Additionally, when stability criteria are applied, sta-ble inverse structures generally have X and Y elementswith similar covalent radii. This observation aligns withthe expectation that X and Y elements should be sim-ilar in size in inverse structures, as their positions areinterchanged compared to regular structures. The regiondefined by χX − χY < 0.15 eV and |rXcov − rYcov| < 0.20Å is highlighted in green, marking a range where struc-tural preferences are distinct. Among the 490 stable in-verse compounds identified, 390 (80%) fall within thisregion. However, the number of regular compounds sat-isfying these criteria is comparable to inverse compounds,indicating that these criteria are robust necessary condi-tions but not sufficient. Conversely, for regular structureprediction, the criteria exhibit high precision (92%) butlower recall (75%).E. Tetragonal distortionHeusler compounds typically crystallize in either thecubic or tetragonal phase. Our HTP computationalstudy revealed that 7,959 X2Y Z-type and 6,909 half-Heusler compounds exhibit tetragonal distortion in theirground state. The primary factor contributing to tetrag-onal distortion is commonly thought to be the peaks inthe density of states (DOS) near the Fermi level (EF ) inthe cubic phase DOS(cubic, EF ) [79]. This conclusionwas supported by the previous high-throughput studieson several hundred X2Y Z compounds containing Fe, Co,and Ni [23, 79–81]. The same conclusion is drawn fromour HTP result spanning significantly broader elementalspace. The distribution of X2Y Z type compounds upto DOS(cubic, EF )=15 eV−1 is shown in Fig.11. Thecounts of cubic and tetragonal Heusler compounds aresorted into bins based on their DOS(cubic,EF ) values.The red curve represents the probability of tetragonaldistortion in each bin. The probability of distortion in-creases as DOS(cubic, EF ) rises. When DOS(cubic, EF )exceeds 3 eV−1, the probability of tetragonal distortionis greater than 80%. The same analysis was extendedto Half-Heusler compounds, and a similar behavior wasfound. When DOS(cubic, EF ) exceeds 3 eV−1, the prob-ability of tetragonal distortion is greater than 70%.It is worth noting that the inclusion of stability criteriaalters the distributions, as shown in the middle and rightpanels of Fig. 11. For half-Heusler compounds, tetrag-onal distortion is observed in 74% of all compounds,but this drops to 31% when only phonon-stable com-pounds are considered. The low ratio in stable com-14−0.2 0.0 0.2Ereg - Einv (eV/atom)020040060080010001200Count(a) All ground states∆E < 0.0; ∆H < 0.3∆E < 0.0; ∆H < 0.3; ωmin = 0−1 0 1χX − χY (eV)−0.50.00.5rX cov−rY cov(Å)reg: 1878inv: 1806reg: 4198inv: 1190(b)All compounds−1 0 1χX − χY (eV)reg: 408inv: 390reg: 1194inv: 100∆E < 0.0; ∆H < 0.3; ωmin = 0−0.10−0.050.000.050.10Ereg−Einv(eV/atom)FIG. 10. (a) Distribution of X2Y Z compounds based on the energy difference between regular and inverse structures (Ereg −Einv). Different stability groups are represented by various colors. (b) Regular/inverse preference of compounds meeting variousstability criteria. The preference is shown by the energy difference (Ereg − Einv), which is color-coded. The electronegativitydifference (χX − χY ) and covalent radius difference (rXcov − rYcov) between X and Y elements are shown on the x- and y-axes,respectively. The region where χX − χY < 0.15 eV and |rXcov − rYcov| < 0.20 Å is marked by green shading.020406080100Percentage(%)oftetra.dist.020406080100Percentage(%)oftetra.dist.0 5 10 1502004006008001000count0 5 10 15X2YZ -type compounds0 5 10 15tetracubic0 5 10 1502004006008001000count0 5 10 15DOS(cubic,EF) (states eV−1 cell−1)Half-type compounds0 5 10 15FIG. 11. Distribution of cubic and tetragonal Heusler com-pounds sorted into bins based on the density of states (DOS)values at the Fermi level in the cubic phase. The red curverepresents the percentage of tetragonal distortion for each bin.The distributions are shown for all compounds (left panels),compounds meeting thermodynamic stability criteria (middlepanels), and compounds additionally satisfying the dynami-cal stability criterion (right panels).pounds indicates that half-Heusler compounds are lesssuitable for applications requiring tetragonal distortion,such as materials with high MCA. Similarly, the oc-currence of tetragonal distortion in X2Y Z compoundsdecreases from 85% in all compounds to 72% amongphonon-stable compounds.V. CONCLUSIONIn this study, we conducted a comprehensive high-throughput stability analysis of 9,072/9,072/324/9,396compositions of regular/inverse/X3Z/half-Heusler com-pounds in both cubic and tetragonal phases, consideringvarious magnetic configurations. In total, 106,235 struc-tures in ground states and metastable states were identi-fied. By applying stability criteria to ground states basedon formation energy, Hull distance, and phonon stabil-ity, we identified 1,898 regular, 1,192 inverse, 81 X3Z,and 840 half-Heusler stable compounds. Among these,1,356 compounds are magnetic, and 631 compounds ex-hibit Tc above 300 K, making them promising candidatesfor further experimental and theoretical exploration ofnew functional materials. Notably, we identified 47 low-moment FiM systems. The spin polarization and anoma-lous Hall/Nernst conductivity were calculated to evalu-ate their potential applications in spintronics and energyharvesting.We validated the Tc calculation method within themean-field approximation against experimental data andfound that a simple correction factor of 0.85 to the Full-potential (FP)-based Tc provides a good agreement withexperimental results. Also, we validated ab initio cal-culation stability criteria alongside experimental datafrom the ICSD, suggesting optimized criteria for achiev-ing higher recall or accuracy, depending on applicationneeds. Our results demonstrated that the inclusion ofphonon stability significantly narrows the pool of viablecandidates, emphasizing its critical role. We expect theseTc calibration methods and refined stability criteria to bevalid to other types of Heusler compounds, such as all-dand quaternary types, as well as similar magnetic sys-tems.Analysis of the HTP data revealed linear relation-ships between Tc and∑i |mi| in 14 Heusler compound15systems, which can be used to do rough estimation ofTc. We also identified correlations between stability andatomic properties, such as atomic radius and ionizationenergy. For regular/inverse preference in X2Y Z com-pounds, our findings aligned with the empirical Burch’srule and further indicated that inverse structures aremore likely when the X and Y elements have similarcovalent radii, although this condition is not sufficientto guarantee an inverse structure formation. In termsof tetragonal distortion, we confirmed a strong correla-tion between tetragonal distortion and a high density ofstates at the Fermi level in the cubic phase inX2Y Z com-pounds, in agreement with the previous work, and iden-tified a similar correlation in half-Heusler compounds.Overall, these insights extend our understanding of sta-bility and structural preferences in Heusler compounds,providing a foundation for more efficient material discov-ery in this and related categories. By refining the criteriafor stability and identifying key atomic correlations, thiswork contributes to the development of next-generationfunctional materials. The comprehensive dataset pro-duced in this work, comprising 106,235 entries of the crys-tal structures, magnetic moments, and the correspondingenergies, 8,180 entries of phonons, 1,356 entries of Tc,and 355 entries of AHC/ANC, is available at [82]. Thisdatabase will be useful not only for querying the stabil-ity of Heuslers but also for developing machine-learningmodels. Development and application of such a machinelearning model based on our database, aiming to exploremore complicated functional Heuslers efficiently, are on-going and will be present elsewhere.DECLARATION OF COMPETING INTERESTThe authors declare that they have no known com-peting financial interests or personal relationships thatcould have appeared to influence the work reported inthis paper.ACKNOWLEDGMENTSThis study was supported by MEXT Program: DataCreation and Utilization-Type Material Research andDevelopment Project (Digital Transformation InitiativeCenter for Magnetic Materials) Grant Number JP-MXP112271550 and as “Program for Promoting Re-searches on the Supercomputer Fugaku” (Data-DrivenResearch Methods Development and Materials Inno-vation Led by Computational Materials Science, JP-MXP1020230327). 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